What should not occur for FMS instrument approaches to be continued past the FAF?

Continuing FMS instrument approaches past the Final Approach Fix (FAF) requires strict adherence to specific conditions to ensure safety and compliance with flight regulations. One critical factor is the presence of an FMS "NO APPR" message displayed on the Primary Flight Display (PFD). This message indicates that the Flight Management System recognizes a problem preventing a proper approach from being conducted. When this message appears, it signals that the system is not capable of executing a stabilized and safe approach, thus making it crucial to not proceed past the FAF under such circumstances.

In contrast, while exceeding maximum airspeed, being under 500 feet AGL, and flying in weather reported below minimums are serious concerns, they may not always directly trigger a critical system warning like the "NO APPR" message does. Rather, these factors might indicate a situation that demands careful consideration or action by the pilot, but they do not represent an immediate system-provided obstacle to continuing the approach through the FAF like the "NO APPR" message does. Therefore, the correct answer reflects a direct system indication that must be adhered to in order to ensure safety during the approach.